Approximation to the stationary distribution of information flows in a communication network

... links in the network The links have finite transmission capacities which are allocated to the information flows concurrently transmitting in the network according to some dynamic bandwidth sharing... results are unsatisfying for providing the benchmark solution to evaluate the accuracy of our approximation algorithm, in absence of analytical solution of the network The reason is that although in. .. processing capacity 1.3 Research contribution As we have mentioned, the data transmitting communication network such as the Internet is once again a hot topic today In recent years, the increasing

Approximation to the stationary distribution of information flows in a communication network

CAO YI (B.Math, Nanjing University)

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

(MANAGEMENT)

DEPARTMENT OF DECISION SCIENCE

NUS BUSINESS SCHOOL NATIONAL UNIVERSITY OF SINGAPORE

Aug, 2005

First and foremost, I would like to take this opportunity to express my sincere appreciation

to my supervisors, Assoc Prof Ou Jihong and Asst Prof Ye Hengqing They duced me into the area of queueing network modelling Their enlightening instruction,plus their comprehensive collection of research materials, allows me to deeply understand

intro-my research topic I am immensely grateful to them for their time and effort on this thesis

in spite of their heavy schedule and other responsibilities Their interest and enthusiasm

in this research topic have made it possible for the completion of this thesis

I am also grateful to A/P Thompson Teo, A/P Chou Cheng Feng, Ms AngChin Teng, Ms Lim May Lin, and of course my supervisors, for the time you sharedwith me in personal conversation and sincere communication, and your greatest support.Lastly, I dedicate this thesis to my parents, who continually encourage and support

me to pursue my goal, who love me most and who I love most

i

1.1 Background 1

1.2 Motivation 4

1.3 Research contribution 6

1.4 Organization of the thesis 8

2 Literature Review 11 2.1 Communication network 11

2.1.1 Bandwidth allocation 12

2.1.2 Stability conditions 13

2.1.3 Stationary distribution 13

2.2 Approximation methods 15

2.2.1 Decomposition method 15

2.2.2 Diffusion approximation 17

2.3 Gibbs sampling method 18

2.4 Chapter summary 19

iii

3 The Communication Network Model 21

3.1 The Network framework 21

3.2 Bandwidth allocation rule 24

3.3 Stationary distribution 27

3.4 Chapter summary 30

4 Approximation Procedure 33 4.1 Modelling processor sharing queues 33

4.2 Approximation algorithm 36

4.3 Chapter Summary 43

5 Gibbs Sampling Method 45 5.1 Generic Gibbs sampling method 46

5.2 Modified Gibbs sampler 49

5.3 Multiple path sampling 50

5.4 Modified Gibbs sampling algorithm 52

5.5 Chapter summary 53

6 Numerical Study 55 6.1 Networks with analytical solution 57

6.1.1 Standard linear network 57

6.1.2 The grid network 63

6.1.3 Single bottleneck 65

6.2 Network without analytical solution 68

6.3 Truncation error 72

6.4 Chapter summary 74

CONTENTS v

7.1 Overview 77

7.2 Summary of findings 77

7.3 Implications of study 78

7.4 Limitations of study 78

7.5 Suggestions for future research 79

In this study we propose a procedure to approximately compute the stationary distribution

of the number of transmitting information flows in a communication network

The flows arrive to the network according to Poisson processes with exponentiallydistributed flow volumes, and traverse through a fixed path of transmission links in thenetwork The links have finite transmission capacities which are allocated to the informa-tion flows concurrently transmitting in the network according to some dynamic bandwidthsharing rule, which ensures the stability of the total number of information flows ongoing

in the network

The procedure is based on dynamic approximation of the bandwidths allocated toconcurrent information flows in the network Numerical examples show that the procedureproduces the numerical solution of the network within 2% of the true values

vii

dis-In recent years, the increasing volume of digital media file transmitting in the networkand the heavy visiting rate to some news websites upon the occurrence of some worldwideevents such as the 911 have deteriorated the previously high performance of the Inter-net, because the current network traffic control mechanism is designed for the small filetransmission situation, rather than today’s demanding situation Therefore the network

is once again put up on the researchers’ table

An abstract framework of this communication network comprises a set of routes

con-1

necting a pair of nodes that are the possible source and destination of information flows,which can be voice conversations in a telephone network, or the digital documents in adata network like the Internet, and a set of transmission links A simplified example ofthis communication network with two links and three routes is illustrated in Figure 1.1,associated with its abstraction in Figure 1.2 Each route carrying an amount of infor-mation flows traverses through a fixed subset of links; while each link has a transmissionbandwidth capacity, which will by some dynamic bandwidth allocation rule, be sharedamong the routes that traverse through it The bandwidth allocated to each route isuniquely determined accordingly, facilitating the transmission of these information flows.

Figure 1.1: Communication Network

When an information flow carrying an amount of data arrives on a given route, aconnection is established on that route After the transmission is finished, the connection

is terminated The same as the traditional queueing network such as the manufacturing

or service network, the communication network can be characterized by the fluctuation

of the number of ongoing connections on each route in the network However, differentfrom a job in a manufacturing job shop or a customer in a service system that visits the

1.1 BACKGROUND 3

Figure 1.2: Communication Network Model

service stations along its route one at a time, an information flow in the communicationnetwork takes up resources simultaneously at all the links along its transmission route

A fundamental issue about this communication network is how to allocate the link’sbandwidth capacity to the routes that traverse through it If we imagine the transmissionprocess on each route as a queueing system, it can be seen that the bandwidth allocationaccording to some bandwidth allocation rules determines the service rate associated witheach queue (route)

An Additive Increase Multiplicative Decrease bandwidth allocation algorithm is plemented in the TCP (the traffic control protocol) of the Internet (see Chiu and Jian

im-1989, Chiu 2000) However, it is observed that the TCP algorithm favors shorter roundtrip time Bertsekas and Gallager (1992) discussed the Max-min fairness bandwidth allo-cation algorithm which intended to maximize the minimum bandwidth allocated to eachroute

Kelly (1997, 1998) proposed the concept of proportional fairness bandwidth allocationand developed a decentralized algorithm to implement it The objective of the allocationrule was to maximize the overall utility of the bandwidth allocations by assuming eachroute had the logarithmic utility function

Mo and Walrand (2000) generalized the above results They proposed a general form

of optimization problem that solved the bandwidth allocations They referred it as the α

proportional fairness allocation The Max-min and Kelly’s proportional fairness

alloca-tions are then the special cases of the α allocation rule.

These fairness bandwidth allocation rules are critical to another important aspect ofthe network, the stability of the network: whether or not the bandwidth allocated toeach route is enough to digest the workload It may be intuitive that the normal offeredload condition is sufficient, that is the total traffic load on each link is within the link’scapacity Unfortunately, Bonald and Massoulie (2000) presented some examples showingthat for some priority bandwidth allocation rules, the condition is insufficient However,many studies show that when the various fairness allocation rules are applied instead, thenormal offered load condition is sufficient See De Veciana et.al (2001)’s discussion for

the max-min fairness allocation, Bonald and Massoulie (2000)’s for the general α fairness

bandwidth allocation, and Ye (2003)’s for the more general utility maximizing bandwidthallocation under general traffic conditions

1.2 Motivation

Although many studies have been conducted for this communication network, compared

to the rich analytical results for the traditional queueing network, little has been availablefor this network Lying at the bottom line of those analytical results is the stationarydistribution of the queueing length of each queue in the network However, even for mosttraditional queueing networks, the analytical solutions are not permitted It adds onextra difficulty for this communication network due to its special bandwidth allocation

1.2 MOTIVATION 5characteristics It is the complexity introduced by the bandwidth allocation rule in thenetwork that precludes to derive the simple closed form solution, e.g a product formsolution, for the stationary distribution of the system In particular, Chiu (2000) showedthe solution is not of product form for a particular network example.

Up to now, only for a few networks with simple structure and certain bandwidthallocation rule, the closed form solutions are derived (see Fayolle et.al 2001) Masoullieand Roberts (1998) showed the closed from solution for the linear network, Bonald andMassoulie (2000) further found the solution for the grid network by solving the same fullbalance equations They suggested that the closed form solution for the network thatviolates the strict underlying assumptions is unavailable

Instead, we may resort to the numerical solution by solving the Markov transition ratematrix However, when the state space is too large, such as a network with too manyroutes, solving the huge matrix is impractical due to the ”curse of dimensionality”

It thus stimulates our interests to design an approximation method to fill the gap,

as what has long been done for those traditional non-product form queueing networks.The underlying idea of our algorithm is to decompose the network into disjoint routes

with each one being represented by an M/M/1 processor sharing (PS) queue The service

capacity is random in that it is dynamically approximated by taking into account theinterdependence of the bandwidth allocations on all the other transmission routes Inparticular, the transmission bandwidth on each route is estimated based on the currentstates of all the other routes in the network The procedure is then iterative: it firstcomputes the marginal distribution of the number of ongoing connections on one route,which provides the base to compute the joint distribution of two routes, etc

The same idea of decomposition approach was developed to approximately compute

the stationary distributions of the traditional queueing network that does not permitthe closed form solution The seminal works include Bitran and Tirupati (1988) andWhitt (1983), and the more recent ones can be found in Whitt (1995) and (1999) The

procedure we construct here is a first attempt to compute (approximately) the stationary

distribution for a queueing network with the simultaneous resource consumption (SRC)characteristics The decomposition approach is modified here in that each isolated queue

is not statically separated from the others but rather dynamically linked in the estimation

of the processing capacity

1.3 Research contribution

As we have mentioned, the data transmitting communication network such as the net is once again a hot topic today In recent years, the increasing volume of digital filetransmitting in the network and the heavy visiting rate to some websites have largelydeteriorated the previously high performance of the Internet, because the current net-work traffic control mechanism is not suitable for today’s demanding situation Thusimprovements are introduced, such as the new bandwidth allocation rules other than theTCP

Inter-Consequently how to evaluate the performance of the network in the context of thesenew improvements becomes an urgent subject But the very few analytical results avail-able up to now is disappointing Although we have the closed form solutions for somesimple networks, there is still no clues how the solution looks like for the general network.Our approximation method is trying to fill the gap It provides a very accurate nu-merical solution to this network Numerical examples indicate that the approximationerror falls within a very small margin of the true solution As another feasible method,

1.3 RESEARCH CONTRIBUTION 7even the most effective modern statistical method, namely the Gibbs Sampling methodunder-performs Thus we can expect that those communication networks with the modestsize could now be solved with a high degree of accuracy To best of our knowledge ourmethod is the first general approximation procedure that provides the numerical solution

to this communication network

Our algorithm has two advantages over those analytical results that are currentlyavailable One is that it is independent of the specific structure of the network in that itcan be applied to any such communication network without adjusting the algorithm toaccommodate its specific structure The network structure is automatically reflected inthe dynamic bandwidth allocation rule, a subfunction in our algorithm

Another advantage is that it is independent of the specific bandwidth allocation rule.The bandwidth allocation rule is packaged in a sub-function and called by the mainfunction in our algorithm This feature is of practical use Because those newly developedbandwidth allocation rules can be tested here in terms of their distinctive impact on thenetwork performance We just modify the subfunction to accommodate the specific rule.There are some practical usages as well For example, in a large network, the accuratesolution of the system is not the first concern The network administrator is concerningwith the bottleneck of the network In this case, we pursue the speed of the solution ratherthan the accuracy by introducing larger truncation error Then the marginal distribution

of each route, which is more accurate than the joint distribution of the system whenthe truncation error is large, will provide information about the dynamics of each route,indicating where the network is in heavy traffic condition and where light traffic Thisresult is definitely not achievable through the inefficient simulation method, or any otherlocal approximation methods that isolate routes for tractability

Given the information of the traffic on each route, we can further adjust the settings

of the algorithm, such that the truncation on each route is treated individually This justment will improve the computational efficiency as well as the accuracy of the solution

ad-of the system

1.4 Organization of the thesis

The organization of this thesis is as follows The following chapter of literature review willprovide the well round background of our study We will first study the communicationnetwork which is the subject of this study The network structure, various bandwidthallocation rules, and the current achievement of some analytical results will be covered.Since the analytical results for the communication network are relatively rare, we willresort to the traditional queueing networks such as the Jackson and BCMP network tosearch for insights from their rich numerical approximation toolbox The most effectiveapproximation methods for the non-product form queueing network will be reviewed.Finally some modern statistical tools developed in the last decade as a very effective way

to compute the complex probability distribution will be briefly introduced In particular,

we will briefly investigate the modern sampling method, namely the Gibbs samplingmethod, which makes computing the complex probability distribution easy by using themodern computational power

Chapter 3 formulates the framework of the communication network under study, anddiscusses the various issues that are critical to the network In Chapter 4, we proposesthe approximation algorithm to compute the solution of the network numerically Themodified Gibbs Sampling which provides an alternative method, other than the ineffec-tive simulation method, to derive the benchmark solution of the network for comparisonpurpose is the subject of Chapter 5 Numerical results are presented in Chapter 6, in

1.4 ORGANIZATION OF THE THESIS 9which we compare the results of our approximation algorithm with those from the Gibbssampling method and the rare analytical results Chapter 7 concludes our study.

situation Therefore the network is once again a hot topic.

2.1.1 Bandwidth allocation

One of the fundamental questions related to improving the network performance in thenew environment is how to allocate each link’s bandwidth capacity among those trans-mission routes that traverse through it, such that the network can effectively handle theworkload on each route

Bertsekas and Gallager (1992) discussed the Max-min bandwidth allocation algorithmwhich intended to maximize the minimum bandwidth allocated to each route such thatthe minimum transmission rate is improved It was later proved that this allocation isthe fairest bandwidth rule (De Veciana et.al 2001)

Kelly (1997, 1998) proposed another, namely the Proportional fairness rule Thisbandwidth allocation maximized the overall utility of the network by assuming a loga-rithmic utility function From the mathematic perspective, Kelly’s study suggested thatthe bandwidth allocation could be obtained by solving an optimization problem, pre-assuming the number of ongoing connections on each route was fixed

Later on, this idea was further developed by Mo and Walrand (2000) They ered a more general optimization problem The corresponding bandwidth allocation was

consid-referred as the α proportional fairness bandwidth allocation rule This allocation rule

includes a wide range of allocation rules, such as the max-min rule, Kelly’s proportional

fairness rule, etc Moreover, a weighting factor w r was introduced into the optimization

problem of the α allocation rule (see Bonald and Massoulie 2000)

Ye (2003) considered a more general bandwidth allocation rule, named the U- utilitymaximizing allocation rule, based on Kelly (2001) and Low (2003)’s work This rulemaximized a more general form utility function aiming to approximate the current TCP

2.1 COMMUNICATION NETWORK 13allocation rule.

2.1.2 Stability conditions

Another important issue is the stability condition of the network, under which the meannumber of ongoing connections on each route will remain finite, not grow into infinite inthe long run Intuitively, it is expected that the normal capacity constrain on each link

is a sufficient condition, which is also referred as the normal offered load condition.Unfortunately, Bonald and Massoulie (2000) showed for some networks with the pri-ority bandwidth allocation rules, this condition is insufficient They concluded that in theabsence of the fairness prerequisite, the bandwidth allocation rules of Pareto efficiency wasnot sufficient to guarantee the stability of the network under the normal traffic condition.According to their suggestion, the stability problem was then studied when somecertain fairness bandwidth allocation rule was applied Some recent results were found inMassoulie and Roberts (1998) for Kelly’s rule, De Veciana et.al (2001) for max-min rule

Bonald and Massoulie (2000) provided the stability results under the general α bandwidth

allocation rule, by using a Fluid model Ye (2003) provided similar stability results for amore general bandwidth allocation, the U-utility maximizing allocation Ye et.al (2003)extended the results to the network with general stationary arrival process

2.1.3 Stationary distribution

Rich studies are conducted in the static context, in that the number of ongoing tions on each route is fixed during the period of study Little has been done to studythe stochastic behavior of the network (Massoulie and Roberts 1998), of which the fun-damental question is the dynamic of the state of the system

connec-In rare cases where the full balance equations are applicable, the system is solved

by using the traditional Markov chain technique For example, Massoulie and Roberts(1998) provided the close form solution for the linear network under Kelly’s Proportionalfairness rule Bonald and Massoulie (2000) extended the above results to Grid network,the generalization of the linear network.

They added on that the analytical result was not available for the more general networkwhere the strict underlying assumptions were not satisfied See Fayolle et.al (2001) forthe similar comments and their study of the approximation method to the star shapednetwork

2.2 APPROXIMATION METHODS 15

2.2 Approximation methods

Long before the fast development of the communication network, the traditional queueingnetwork has been extensively studied since Jackson’s seminar work (Jackson 1957,1963),and later the BCMP theory (Baskett et.al 1975) These networks have an attractiveproperty that the stationary joint distribution of the system could be explicitly expressed

in a product form But in more general cases where the local balance equations arenot available, most queueing networks do not permit the product form solution Only

by approximation methods can we obtain an approximated solution Among them, themost effective approximation method, namely the decomposition method, borrowed theunderlying idea of Jackson’s product form solution

2.2.1 Decomposition method

Although the queueing network is difficult to analyze in a whole, it can be divided intoseveral small subnetworks, in the extreme case each subnetwork consisting of only onequeue Then each subnetwork is analyzed individually Finally by taking into accountthe interaction between the different subnetworks, the individual results are combinedtogether to obtain the approximated solution to the entire network

Based on this idea, this method is widely used when the queues of the network can

be divided into weakly interrelated groups The advantage of this method is that itrequires little on the computational time which is independent of the size of the entirenetwork While the disadvantage is that the uncertainty of the accuracy level of thesolution remains, and the convergence of the solution is not guaranteed (Gelenbe andPujolle 1987)

As noted by Harrison and Petal (1993), the decomposition method gave a very

accu-rate approximation when the system was almost of product form It also gave remarkablygood results even when many product form assumptions were violated.

The most often used decomposition method is found in Kuehn (1979), later in tran and Dasu (1990), and is especially applied to the open queueing network It firstdecomposes the network into a set of single queues, and then analyzes the effective inputand output process of each queue The interaction among the separated queues is re-flected by square of coefficient of variations of arrival processes at each queue (see Whitt1983,1984’s a series of superior work and the reference therein)

Bi-With some normal assumptions, the method consists of three steps: flow aggregating(see Bitran and Tirupati 1988, Whitt 1982,1983), flow analysis (see Pujolle and Ai 1986),flow splitting (see Whitt 1984, Disney and Konig 1985) Finally by combining the threesteps, we obtain the system of linear equations (see Bitran and Tirupati 1988) that solvethe effective arrival rates and their interrelationship, which are used to compute thevarious performance measures of the system (see Albin 1984, Bitran and Dasu 1990).Whitt (1983) developed a software, namely the QNA (Queueing Network Analyzer)

to implement the above procedures The advantage of the QNA over other similar solverssuch as the PANACEA is that it requires only renewal arrivals rather than Poisson arrivals

as in the other solvers, thus the modelling error is largely eliminated The drawback is that

it assumes un-correlated and un- autocorrelated arrivals, thus may encounters difficulties

in the heavy traffic bottleneck situation (see Kim et.al 2000, Suresh and Whitt 1990,and Whitt 1995)

Bitran and Tirupati (1988) considered the decomposition method for the multipleproduct network with deterministic routing, where the interaction among the differenttypes of product streams is a concern in the splitting step They proposed a way to take

2.2 APPROXIMATION METHODS 17into account this interaction.

2.2.2 Diffusion approximation

The decomposition method is successful for the queueing network with normally discretearrivals When the arrivals are intensive, each increase or decrease of the populationcomparing to the total population is relatively small Thus it is suggested by Harrison(1985), Reiman (1984) to model the population as a Brownian Motion Because of itssimilarity to the diffusion equations for the ideal gas, this approximation method is aswell called the diffusion approximation

It was shown (Reiman 1984) that under the heavy traffic condition, the J- dimension queue length process associating with a certain type of open J- dimension queueing net-

work, when properly normalized, converges to a corresponding reflected Brownian Motion(RBM) with drift

Many studies have been done to identify the underlying RBM model of the singleclass queueing network, and to convert parameters of the network to the inputs of thecorresponding RBM model See Harrison and Williams (1987) and Harrison et.al (1990)for the survey of work on open and close queueing network respectively

The justification of this approximation method is based on the ”heavy traffic limittheorem” For example, Dai and Dai (1999) proved the theorem be valid for the finitebuffer single class queueing network

For two dimensional RBMs, the analytical solutions were derived in Harrison et.al(1985), Foddy (1983) In higher dimension, RBMs with exponential form solutions wereidentified in Harrison and Williams (1987), Williams (1987) In general, we have to iden-tify and then solve a set of Partial Differential Equations (PDEs) to obtain the numericalsolution

Dai and his colleagues have done a lot of work on identifying a unique set of PDEs, andexploring efficient methods to solve the RBM model, see Dai and Harrison (1991,1992),Dai et.al (1994)’ SBD method, Harrison and Nguyen (1990)’s QNET, Chen et.al (2002)’susing of the finite element method for the finite buffer network Recent applications ofthe RBM model to multiple product queueing network can be found in Chen et.al (2001).

2.3 Gibbs sampling method

In our experiments, we found that the simulation results are unsatisfying for providingthe benchmark solution to evaluate the accuracy of our approximation algorithm, inabsence of analytical solution of the network The reason is that although in practice,the simulation method is most of time used to provide the benchmark of mean queuelength, in our case the solution of the system is a set of huge number of values Thus toobtain an accurate solution by conducting the naive simulation method is not satisfying

In order to quantify the accuracy of our approximation algorithm, we resort to the moreeffective modern statistical method, in particular the Gibbs sampling method (Germenand German 1984, Liu 1996), which provides the benchmark solution

Gibbs sampling was first developed by Geman and Geman (1984) for simulating terior distribution in image reconstruction As a family member of the modern samplingmethods (the Monte Carlo Markov Chain technique), it gives a way to approximate theprobability distributions through sampling In particular, it gives a convenient way tosample from a complex distribution

pos-The Gibbs technique can be theoretically justified by the Monte Carlo Markov Chain(MCMC) theory, and it does do an excellent job in assisting statisticians to compute theposterior marginal distribution efficiently and accurately For the complicated applica-

2.4 CHAPTER SUMMARY 19tions and additional references, see Casella and George (1992), Gelfand et.al (1990), andGelfand and Smith (1990) and references therein.

For the case of discrete sample space, Liu (1996) modified the naive Gibbs sampler, andproved that the modified Gibbs sampler was statistically more efficient than the randomscan Gibbs sampler (a type of the naive Gibbs sampler) It was essentially a randomsampler, which updated the sample in each cycle with an acceptance probability, as that

of the MCMC sampler

2.4 Chapter summary

This chapter reviews a rich collection of the literatures relative to our study For thecommunication network that once again becomes a hot topic in recent years, the band-width allocation rule, stability condition, and the stationary solution are among the mostimportant issues of theoretical and practical interests At present time, the bandwidthallocation issue is under extensive studies, but the stationary solution remains to be achallenging problem

To search for the approximation method to compute that stationary solution which issimilar to that of the the traditional queueing network, classic Jackson network providesthe theoretical insights to our understanding of a network system Although it fails to fitinto more realistic network, its product form solution does suggest the availability of thedecomposition approximation method for a wide range of the traditional networks Up

to now, the decomposition method still dominates the approximation area of most of reallife networks due to its relatively easy formation In the parallel side, reflected Brownianmotion approximation is used to approximate the solution of networks with heavy trafficfeature Modern statistical method such as the MCMC and Gibbs sampling methods

provides a new angle to look at the difficult problem.

Chapter 3

The Communication Network Model

In this chapter, we will introduce the communication network model, which is often used

to model today’s data transmission network such as the Internet, WAN, LAN, etc Thefundamental issues about this network will be covered, including the bandwidth allocationrule, the stability condition, and some analytical results available up to date

3.1 The Network framework

The communication network comprises a set of L transmission links, which provide the

bandwidth for the information flows transmitting on the network; and a set of routes

r’s carrying information flows, with each one be a non- empty subset of L, in the sense

that each route r traverses a set of links l’s Denote the set of all possible routes as R such that r ∈ R, with a total of M routes Conversely, let a fixed 0 − 1 incidence matrix

A = (A lr , l ∈ L, r ∈ R) indicate which links are in a particular route, and let R(l) indicate

all the routes that have link l on their path.

A simple abstraction of a real-life communication network (linear network) is trated in Figure 3.1 This network consists of two links and three routes, with route 1

illus-21

and 2 traverse through link 1, 2 respectively, and route 3 through both link 1 and 2.

Figure 3.1: Linear network model

Another basic network is an extension of the linear network, the grid network, Figure

3.2 It consists of several horizontal routes r k and vertical routes r l, (in the linear networkcase, there are only one horizontal route: the longest route traverses all links)

Figure 3.2: Grid network model

The third basic network most often studied is the cyclic network in Figure 3.3 Inthis simple cyclic network, it consists of 6 links and 6 routes, with each route traverses 3links in a symmetric fashion

Information flows arrive to route r according to a Poisson process with rate λ r; andthe flow’s volume (i.e the size of the file that will be transmitted) is an iid exponential

random variable with mean v −1

r On each route, an arriving information flow will be

3.1 THE NETWORK FRAMEWORK 23

Figure 3.3: Cyclic network model

immediately transmitted through the link, no matter whether there is any other flowsbeing transmitting In other words, the flows are transmitted simultaneously upon their

arrival The above transmission mechanism can be well modelled by a traditional M/M/1

Processor Sharing (PS) queue in that the service capacity (the bandwidth here) is equallyshared among the present ongoing information flows on that route, with the bandwidth

Λr being determined by the bandwidth allocation rule (defined later)

Note that the difference of this type of communication network with the traditional

queueing network lies only in that the transmitting information flows in route 3

simul-taneously consume the link capacity on both link 1 and 2 which lie on the path of that

route In other words, every bit of the flow that has been transmitted through link 1 willimmediately goes to link 2 and be transmitted as if the link 1 and 2 are seamlessly linked;rather than being transmitted by only one of the link 1 and 2 at a time, like a customerbeing served by two bank tells one by one Thus the bandwidth for route 3 is in effectrestricted to be the minimum of bandwidth given by link 1 and 2

The number of information flows being transmitting on each route fluctuates It

increases when a new flow arrives, decreases when a flow is completely transmitted throughthe link(s) on the its path Same as for the traditional queueing network, one of thefundamental questions concerning to evaluating the network performance is the stationarydistribution of the number of the ongoing transmitting flows in the network (the analogy to

the number of queueing customers in a queueing network) At time t > 0, let n r (t) denote the number of flows that are currently transmitting on route r, and n(t) = {n r (t) : r ∈ R}

be the vector of the numbers of ongoing flows in the network We are interested in the

probability distribution of the n(t) in the long run.

3.2 Bandwidth allocation rule

We now consider a remaining issue that completes the construction of the network, that

is how the link allocates its bandwidth capacity to each route that traverses through it

Each link l ∈ L has a bandwidth capacity C l > 0, that is the volume of information

flow that can be transmitted through the link per unit of time The bandwidth capacity

of each link is allocated to the routes that traverse through this link according to some

dynamic bandwidth allocation rules in the sense that the rule re-adjusts the allocation as

the number of transmitting information flows on each route fluctuates

In particular, let Λ(n(t)) = {Λ r (n(t)) : r ∈ R} denote the bandwidth allocated to each

route, determined by the generic bandwidth allocation rule, with Λr (n(t)) be the amount

of bandwidth allocated to route r at time t when the numbers of transmitting flows on all routes in the network are n(t) = {n r (t), r ∈ R} Here we implicitly assume the allocation rule depends only on the number of transmitting flows on each route at time t, which may

ignore some realistic consideration, but is widely adopted in the present literatures.People have developed some bandwidth allocation rules that achieve these two goals as

3.2 BANDWIDTH ALLOCATION RULE 25much as possible, such as the max-min fairness rule (Bertsekas and Gallager 1992), whichmaximizes the minimum bandwidth allocated to each route such that to some extent theminimum service rate is guaranteed.

Later on Kelly proposed another fairness rule, the proportional fairness rule, thatmathematically corresponds to an optimization model that determines the bandwidthallocations Now almost all bandwidth allocation could be determined by solving this op-timization model as extended by Mo and Wolrand (2000), including some rules developedafter Kelly’s rule That optimization model reads:

r∈R

U(n r , Λ r)subject to X

r∈R(l)

Λr ≤ C l for l ∈ L

Λr = 0, if n r = 0Kelly’s proportional fairness rule corresponds to the logarithm utility maximization func-tion

Given n r’s , the bandwidth allocation rule based on this optimization model is aimed

to maximize the overall utility of the network The constrain simply states that thebandwidth allocations cannot exceed the link’s capacity Therefore the specific forms ofthe utility functions differentiate those various bandwidth allocation rules For exampleKelly’s proportional fairness rule maximizes the following utility function:

r∈R

n r log(Λ r /n r)That reads the rule maximizes the total benefit all over the flows in the network byassuming each flow possess logarithm utility upon the bandwidth it shares

Mo and Walrand (2000) developed a very general utility function, the α- proportional

fairness rule, that includes many specific cases we have mentioned It takes the form as

It should be mentioned here that the determination of the bandwidth for each route

by solving the optimization problem is conducted in a static context in that the number

of currently transmitting flows on each route is pre-fixed Although it is hard to believethis scheme could really be implemented in a realistic network, it does not impair thefundamental study of the network (see Ye et.al 2003) Thus to study the dynamic behavior

of the network (that is the fluctuation of the number of transmitting flows), we take thismechanism as granted, that is the bandwidth allocation is immediately determined byre-solving the optimization problem once the number of transmitting flows on any routefluctuates, and remains unchange until the next fluctuation occurs

For the three basic networks we have presented before, we can obtain their weighted

α proportional fairness bandwidth allocation in a close form by solving the optimization

problem These three network cases will be used to test our approximation algorithmlater in the chapter of numerical study, thus it is worth here to derive their bandwidthallocations respectively (Note that for Kelly’s proportional fairness rule, it is simply that

w i = 1, α = 1)

Here we list the bandwidth allocation only for the linear network with unit capacity

links (that is C l = 1), (more on the bandwidth allocations for the grid, cyclic network

later) The weighted α proportional fairness bandwidth are:

Λ0: bandwidth rate allocated to the go-through-all-links route R0

Λr : bandwidth rate allocated to each go-through-one-link route R r

n r : number of transmitting information flows on route R r

w r : weighting factor for route R r

For the grid network as well with unit capacity links, the α proportional fairness

bandwidth rates are derived in the same way:

with n k be the number of transmitting flows on the horizontal route R k , and n l be that

on the vertical route R l

For the cyclic network, which consists of 2L links and 2L routes of length L, and route

l crosses link l + 1, , l + L, the α proportional fairness allocation is:

Up to now most of studies are focused on the static analysis of the network, that is

to assume the number of transmitting flows on each route is fixed during the period of

study, such as the determination of the bandwidth allocation The dynamic behavior ofthe network is not yet studied What if the state (the number of transmitting flows oneach route) fluctuates? How does this fluctuation affect the network performance?

As the same for the traditional queueing system, the state of the system fluctuates withthe time Given the stability condition Pr∈R(l) ρ r < C l, which guarantees the long runstationary status of the network exists under the various fairness bandwidth allocationrules, we are interested in the long run behavior of the system, that is the stationarydistribution of these states

Continuous time Markov chain technique is still the root to derive any exact solution

of a queueing system other than various approximation methods The communicationnetwork studied here is also the case

By assuming Poisson arrivals on each route, and iid exponential flow volume, the state

of the system can be modelled as a continuous time Markov chain as follows:

The state space is well defined, that is ~n(t) = {n r (t), r ∈ R} The transition rates

depend on the state through the bandwidth allocation rule (Λr (n)), as well as on the arrival rates (λ r ), mean flow volume (µ −1

where Λr (~n) is the bandwidth rate allocated to route R r, depending on the states of the

system (~n), and e r is a vector having the same dimension with ~n(t), with 1 at the r-th

position and 0 for all the others

For the linear network under Kelly’s proportional fairness allocation rule, the transition

i n i is the bandwidth rate allocated to the longest route R0, i.e Λ0 in the

linear network case, where α = 1, w i = 1 (see section 3.2)

For the grid network with the same settings, let the state be {(x k , y l )}, representing

the pair of horizontal and vertical routes, the transition rates are then:

It is then proved by Massoulie and Roberts (1998) that for the linear network with

unit capacity link (C l = 1) and under Kelly’s proportional fairness rule, if the normal

offered load condition is satisfied, that is: Max 1≤i≤L ρ0+ ρ i < 1, where ρ i = λ i /µ i is the

traffic load on route i, then the process n(t) is reversible, and has stationary distribution:

ρ n i

where C = QL (1 − ρ0)L−1

1(1 − ρ0− ρ i)

where C is the normalization constant.

For the grid network, the similar result could be derived by solving the balance

equa-tions of the continuous time Markov chain The Markov process (x k , y l) is reversible, andwith the stationary distribution:

ρ x k k

Y

ρ y l l

Even trying to derive the numerical solution by solving the Markov transition ratematrix is impractical One of the reasons is that the size of the transition rate matrixincreases exponentially with the number of routes and number of states For example,

for a simple linear network with K routes, and let the state (the number of transmitting flows) on each route be the same: {0, 1, , N − 1} Thus the state of the system as a whole (that is to consider the routes jointly) is a K dimensional vector (n1, , n K) with

n i = 0, 1, , N − 1 The size of the transition rate matrix is then N K by N K When N and K are too large, it far exceeds the capability of the computer to solve the matrix.

3.4 Chapter summary

In this chapter, we formulate the framework of the communication network under study

It consists of a set of routes and links, on which the very important bandwidth allocationsare determined by solving an optimization problem Next the state of the system, namelythe number of transmitting flows on each route, is modelled by a continuous time Markovchain, with the transition rates be determined by the arrival rates and bandwidth alloca-tions exclusively By solving the balance equations of the Markov chain, the closed formsolution to the stationary distribution of the state of the system is derived for only a few